Recently whenever I play a GameCube backup from DIOS MIOS 2.2 (with a USB drive), sometimes the Wii shuts down. This happens in the middle of gameplay (that means I can get past the splash screen), or whenever the Wii needs to access the USB drive.
This happened with two games:
Doubutsu no Mori e+ (JAP)
Fire Emblem: Path of Radiance (USA)

My USB stick is an 8GB SanDisk Cruzer. I use it to backup both GCN and Wii games. It can run Wii games perfectly fine, but I'm having problems with GCN games.

I've come to think that DM and GC games have "spikes" in data streaming and if the medium (usb HDD, thumbdrive, etc.) isn't capable of "keeping up", the system just shut down.

So if you don't wanna have this problem try to use
- "fast" drives (HDDs or SSDs, not cheap USB thumbdrives by any mean, and that's probably the problem of the OP)
- good quality USB cables (and as short as possible)
- NO USB EXTENSION CABLES (that's how I first experienced and then solved this problem)

As further evidence, in this other thread (one of many) about the issue